52–54. Force on a window A diving pool that is 4 m deep and full of water has a viewing window on one of its vertical walls. Find the force on the following windows. 


The window is circular, with a radius of 0.5 m, tangent to the bottom of the pool.

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = 1−x²,x = 0, and y = 0, in the first quadrant; about the y-axis

Find the area of the region described in the following exercises.

The region in the first quadrant bounded by y=x^2/3 and y=4

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

x =2

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.

ρ(x) = {1 if 0≤x≤2 {2 if 2<x≤3

Suppose f and g have continuous derivatives on an interval [a, b]. Prove that if f(a)=g(a) and f(b)=g(b), then ∫a^b f′(x) dx = ∫a^b g′(x) dx.

Find the area of the surface generated when the given curve is revolved about the given axis.

y=8√x, for 9≤x≤20; about the x-axis